Ontario 09 Academic (MPM1D)
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Finding linear equations in context
Lesson

Now that we know how 

  • to graph linear relationships
  • to find the equations of linear functions
  • to use algebra and graphs to extract information
  • to find intercepts and constant values, and
  • that the slope of a linear function represents constant change.

we can put this to use to solve a range of real life applications.

It's all the same mathematics, but this time you will have a context to apply it to. 

Some examples will be the best way to show you how the mathematics we have learnt can be applied to everyday situations.  

Examples

Question 1

A carpenter charges a callout fee of $\$150$$150 plus $\$45$$45 per hour.

  1. Write an equation to represent the total amount charged, $y$y, by the carpenter as a function of the number of hours worked, $x$x.

  2. What is the slope of the function?

  3. What does this slope represent?

    The total amount charged increases by $\$45$$45 for each additional hour of work.

    A

    The minimum amount charged by the carpenter.

    B

    The total amount charged increases by $\$1$$1 for each additional $45$45 hours of work.

    C

    The total amount charged for $0$0 hours of work.

    D

    The total amount charged increases by $\$45$$45 for each additional hour of work.

    A

    The minimum amount charged by the carpenter.

    B

    The total amount charged increases by $\$1$$1 for each additional $45$45 hours of work.

    C

    The total amount charged for $0$0 hours of work.

    D
  4. What is the value of the $y$y-intercept?

  5. What does this $y$y-intercept represent?

    Select all that apply.

    The total amount charged increases by $\$150$$150 for each additional hour of work.

    A

    The maximum amount charged by the carpenter.

    B

    The callout fee.

    C

    The minimum amount charged by the carpenter.

    D

    The total amount charged increases by $\$150$$150 for each additional hour of work.

    A

    The maximum amount charged by the carpenter.

    B

    The callout fee.

    C

    The minimum amount charged by the carpenter.

    D
  6. Find the total amount charged by the carpenter for $6$6 hours of work.

Question 2

The table shows the linear relationship between the length of a mobile phone call and the cost of the call.

Length of call (minutes) $1$1 $2$2 $3$3
Cost (dollars) $7.6$7.6 $14.4$14.4 $21.2$21.2
  1. Write an equation to represent the cost of a call, $y$y, as a function of the length of the call, $x$x.

  2. What is the slope of the function?

  3. What does the slope tell you?

    The connection fee

    A

    The cost of each additional minute

    B

    The cost of the phone

    C

    The cost of a $1$1-minute call

    D

    The connection fee

    A

    The cost of each additional minute

    B

    The cost of the phone

    C

    The cost of a $1$1-minute call

    D
  4. What is the $y$y-intercept?

  5. What does this $y$y-intercept tell you?

    The cost of each additional minute

    A

    The cost of the phone

    B

    The cost of a $1$1-minute call

    C

    The connection fee

    D

    The cost of each additional minute

    A

    The cost of the phone

    B

    The cost of a $1$1-minute call

    C

    The connection fee

    D
  6. Find the cost of a $6$6-minute call.

Question 3

The graph shows the amount of water remaining in a bucket that was initially full before a hole was made in it's side.

Loading Graph...

  1. What is the slope of the function?

  2. What is the $y$y-value of the $y$y-intercept?

  3. Write an equation to represent the amount of water remaining in the bucket, $y$y, as a function of time, $x$x.

  4. What does the slope tell you?

    The amount of water remaining in the bucket after $2$2 minutes.

    A

    The amount of water that is flowing out of the hole every minute.

    B

    The time it takes for the bucket to be completely empty.

    C

    The time it takes the amount of water remaining in the bucket to drop by one litre.

    D

    The amount of water remaining in the bucket after $2$2 minutes.

    A

    The amount of water that is flowing out of the hole every minute.

    B

    The time it takes for the bucket to be completely empty.

    C

    The time it takes the amount of water remaining in the bucket to drop by one litre.

    D
  5. What does the $y$y-intercept tell you?

    The capacity of the bucket.

    A

    The amount of water remaining in the bucket after $30$30 minutes.

    B

    The amount of water remaining in the bucket when it is empty.

    C

    The size of the hole.

    D

    The capacity of the bucket.

    A

    The amount of water remaining in the bucket after $30$30 minutes.

    B

    The amount of water remaining in the bucket when it is empty.

    C

    The size of the hole.

    D
  6. Find the amount of water remaining in the bucket after $54$54 minutes.

 

 

 

 

 

Outcomes

9D.AG3.03

Describe the meaning of the slope and y-intercept for a linear relation arising from a realistic situation and describe a situation that could be modelled by a given linear equation

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